class: center, middle, inverse, title-slide # Lecture 4 ## Statistical Models and Notation ### Psych 10 C ### University of California, Irvine ### 04/06/2022 --- ## Models in Science - Models are abstract representations of the world. -- - They are a way in which we can formalize our believes about events. -- - For example, if we have an experiment where we throw a coin and have two two competing ideas about the coin: -- - The coin is **fair**. -- - The coin is **not fair**. -- - We can formalize these two believes using a statistical model. -- - The coin is fair: `\(P(\{heads\})\ =\ P(\{tails\})\ =\ 0.5\)` -- - The coin is not fair: `\(P(\{heads\})\ \neq\ P(\{tails\})\)` -- - We moved from two verbal statements about our believes regarding the coin to two formal statements bout the proabibility of "heads" as an outcome. --- ## Statistical Models - Statistical models are the formal representation of our believes or hypothesis about the outcomes of an experiment. -- - Given that we assume that the outcomes are probabilistic, our models will have a probabilisitic component associated with them. -- - Given the probabilistic nature of our observations it will be almost imposible for us to tell if a model is TRUE or FALSE. However, we can compare how useful they are on a given situation. -- - Statistical models will allow us to make predictions about our observations, ans we will use these predictions to compare how useful they are. -- - However, before we continue it will be useful to introduce some notation! -- - This will provide us with a way to express our models in a formal and standard way. --- class: inverse, middle, center # Notation --- ## Example: Memory Experiment - Whenever we run an experiment, we have a variable that we are interested in. In homework one we had the results of an IQ test, in our memory example we had the number of words that are correctly recognized. -- - We will call the variable that we are interested in the **dependent** variable. -- - To reference a particular observation in an experiment we will use the lowercase letter `\(y_{i}\)`. -- - Where the subscripts `\(i\)` indicates our *i-th* observation. -- - Let's look back at the data for test one in our memory experiment --- ## Example - Now we will use the data only from one of the test of our experiment, however, we have added a new variable that indicates the **age_group**. <div id="htmlwidget-180f5d00873fa1ca8322" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-180f5d00873fa1ca8322">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100"],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100],[20,29,29,25,23,26,23,24,21,31,27,29,21,26,25,19,27,21,23,17,21,10,21,27,21,26,18,26,22,24,23,22,28,25,27,22,15,22,26,29,22,24,21,20,24,22,17,21,17,34,64,43,51,49,50,55,53,62,41,47,52,65,52,37,53,60,46,46,54,55,48,44,56,40,49,50,48,52,40,47,50,40,45,56,51,49,57,54,45,42,50,49,40,56,53,51,58,37,60,53],[46,49,48,44,45,44,48,42,42,44,46,41,47,46,46,47,47,44,42,48,47,48,47,45,44,45,45,46,49,44,49,46,47,45,48,43,48,48,43,48,47,47,42,46,43,46,46,45,46,43,43,43,46,46,42,45,46,43,48,44,45,45,47,43,45,42,44,42,42,42,40,46,45,47,43,44,46,43,46,43,42,45,47,46,41,45,46,44,44,43,44,47,47,41,45,43,45,47,41,43],[300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300,300],["test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1","test_1"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>age<\/th>\n <th>correct<\/th>\n <th>time_test<\/th>\n <th>test_id<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ## Example - In our experiment we have `\(100\)` observations, in other words we have a sequence of `\(y\)` values that goes from `\(y_1, y_2, \dots, y_{100}\)`. -- - Where `\(y_1\)` denotes our first observation, in other words, `\(y_1 =\)` 46. -- - `\(y_{50}\)` denotes our *50-th* observation or `\(y_{50} =\)` 43. -- - When we have more than one group we can add a second subscript to our observations. -- - For example, let's divide our memory data example in two groups, participants who are 35 or younger will be classified as "young" while participants that are older than 35 will be classified as "elder". ```r memory <- memory %>% mutate("age_group" = ifelse(test = age > 35, yes = "elder", no = "young")) ``` --- ## Example - Now we have two groups and we can denote their observations using two subscripts -- - For example, the *i-th* observation of the "young" group can be denoted with `\(y_{i1}\)` while the *i-th* observation of the "elder" group will be `\(y_{i2}\)`. -- - In general, we can say that the *i-th* observation of the *j-th* group will be `\(y_{ij}\)` -- - In our experiment, the first observation (number of correctly recognized words) of the first participant in the "young" group was `\(y_{11}=\)` 46. -- - While the first observation of the "elder" group was `\(y_{12}=\)` 43 --- ## Statistical models